{"id":"51a44aaf-7343-493c-a0b0-4ad35e7e0b1c","arxiv_id":"2412.03810","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A scale-independent pairwise-force smoothed particle hydrodynamics model with new polynomial force profiles is calibrated and validated for computing sessile droplet shapes on complex substrates.","lead":"The authors propose a new particle-based simulation method for computing the shapes of liquid droplets resting on complex surfaces such as rough, patterned, or plant-leaf substrates. The method is validated against known physics and applied to demonstrate wetting transitions and droplet shapes on a virtual wheat leaf.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Contact-angle calibration rests on a density isosurface that demonstrably loses more than 10% of droplet volume; the fluid-only isosurface (Eq. 12) is inconsistent with the simulation density that includes solid particles, so the s_fs/s_ff–theta_CA curve may be biased.","rationale":"The reader identifies the isosurface/contact-line definition as the weakest assumption, and my analysis converges on the same point but sharpens it with a concrete mechanism: the density field used for the isosurface (fluid particles only) is not the density field computed in the simulation (fluid plus solid particles in Eq. 3). This inconsistency is located precisely in Section 2.2.4 versus Section 3.3, and it directly threatens the calibration that everything else uses. The surface-tension part of the claim is independently supported by the Laplace-pressure and oscillating-droplet tests, and the pillared-substrate case is a plausible qualitative benchmark, so those parts are not the main risk. The contact-angle calibration, however, is the linchpin for the predictive claim: if the isosurface is biased near the substrate, the s_fs/s_ff–theta_CA relationship in Figures 10-11 is not reliable, and the hydrophilic-stripe angle map and wheat-leaf contact line inherit the same error. The 10% volume filter makes the risk concrete rather than hypothetical, because it shows the isosurface can be seriously wrong in some simulations; excluding those cases does not guarantee the remaining ones are unbiased. The proposed test is a single computational experiment that directly compares the fluid-only isosurface with the all-particle density field, and would settle whether the calibration is robust. I therefore keep the reader's CONDITIONAL verdict unchanged: the paper should not be rejected, but the predictive contact-angle claim should be conditioned on this check or an equivalent demonstration of isosurface robustness.","tokens_in":19294,"tokens_out":4139,"duration_ms":48057,"concrete_test":"For a representative flat-surface calibration case in the mid-range (e.g., s_fs/s_ff ≈ 2), reconstruct the half-density isosurface two ways: (i) with fluid particles only as in Eq. (12), and (ii) with the all-particle density actually used in the continuity equation (Eq. 3), including solid boundary particles. For each isosurface, recompute the enclosed volume, the contact line at distance 2Δx from the substrate, and the Young-Laplace best-fit contact angle. If the fitted theta_CA or the contact-line position changes by more than the fitting tolerance (or by more than about 5 degrees), the current calibration and all contact-line visualizations are biased, and the calibration should be redone with a consistent density definition or an explicit isosurface correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the model predicts contact angles and complex contact lines from s_fs/s_ff depends on the assertion that the half-density isosurface (Eq. 12) faithfully locates the liquid-gas interface, and that the contact line is where this isosurface is 2Δx from the substrate. That assumption is insecure for two reasons. First, the SPH density actually evolved in the simulation includes fixed solid particles in the sums of Eq. (3) (Section 2.2.4), whereas the isosurface in Eq. (12) sums only fluid particles. Near the substrate the fluid-only sum is truncated, so the half-density surface is displaced relative to the density field that governs the forces; the apparent contact line is shifted by an O(Δx) offset that is not physically calibrated. Second, the volume-exclusion filter in Figures 10 and 11 discards any droplet whose isosurface encloses a volume differing from the true droplet volume by more than 10%. This is direct evidence that the isosurface can be significantly wrong, and the filter does not establish that the surviving cases are unbiased; if the excluded cases are correlated with s_fs/s_ff, the published calibration curve is biased. Since the same isosurface and distance criterion are used to measure the contact-angle transition in Figure 14 and the wheat-leaf contact line in Figure 16, any systematic isosurface bias near the substrate propagates into every case-study conclusion, not just the calibration.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a pairwise-force smoothed particle hydrodynamics (PF-SPH) model with new polynomial force profiles (Eq. 8) and an H-scaled interaction (Eq. 7) intended to make simulated surface tension and contact angle scale-independent. The model is calibrated by measuring surface tension from Laplace pressure in static spherical droplets and from ellipsoidal droplet oscillation frequencies, both giving a linear relation sigma = 30.96 s_ff. The contact angle is calibrated by fitting semi-analytical Young-Laplace profiles to density isosurfaces of sessile droplets and plotting the fitted angle against s_fs/s_ff. The method is then demonstrated on a pillared superhydrophobic-type surface, a chemically patterned surface with a hydrophilic stripe, and a reconstructed wheat leaf, with the contact line obtained without explicit tracking.","tokens_in":19666,"tokens_out":4679,"duration_ms":48441,"significance":"If the calibration is reliable, the paper makes a useful contribution: the linear sigma-s_ff relation is corroborated by two independent tests, the H-scaling addresses a known resolution dependence in PF-SPH, and the case studies show the model can handle complex substrates without explicit contact-line tracking. The open-source implementation is a further strength. However, the contact-angle calibration is partly self-referential and rests on an isosurface definition whose accuracy near the substrate is not established. These issues affect the central predictive claim, so the contribution is not yet fully supported, although the underlying modelling idea is promising and the Laplace-pressure and oscillation results provide a solid core.","major_comments":[{"comment":"The contact-angle calibration is partly circular: theta_CA is defined as the best-fit parameter of a Young-Laplace profile fitted to the simulated isosurface, and the same fitted values are then presented as agreement with Young-Laplace. This establishes that the simulated shapes are Young-Laplace-like, but it does not independently validate the model's wetting behaviour. Please provide an independent measurement of the contact angle (for example, a tangent construction at the contact line, or comparison with an independent numerical method) or explicitly reframe the claim as calibration rather than validation.","section":"Section 3.3, Eq. (13), Figures 10-11"},{"comment":"The interface in Eq. (12) is defined from a fluid-only density sum, whereas the density evolved in Eq. (3) and used in the pressure and pairwise forces includes fixed solid particles. Near the substrate the fluid-only sum is truncated, so the half-density isosurface may be displaced by an O(Delta x) offset relative to the density field that actually controls the forces. Since the contact line is defined as the intersection of this isosurface with the 2 Delta x offset surface (Section 4.3), any systematic isosurface bias propagates into the calibration curve and the wheat-leaf contact line. Please test the sensitivity of the calibration to an alternative interface definition based on the full density field, or demonstrate convergence with resolution.","section":"Section 2.2.4 versus Eq. (12)"},{"comment":"Droplets are excluded when the isosurface-enclosed volume differs from the true droplet volume by more than 10%, which is direct evidence that the isosurface can be inaccurate. The filter may bias the s_fs/s_ff-theta_CA curve if exclusions correlate with the force ratio. Please report the number and parameter values of excluded cases and show that the calibration is unchanged when the threshold is varied or when a corrected interface is used.","section":"Figures 10 and 11"},{"comment":"The contact angle along the contact line is measured using the surface normal of the isosurface at a fixed height H/2 above the substrate, rather than at the contact line itself. This offset is not physically calibrated and may depend on resolution; it should be justified or replaced by a definition tied to the actual contact line, especially because the transition plot is used to support the chemically patterned surface claim.","section":"Section 4.2, Figure 14"}],"minor_comments":[{"comment":"The claim that sigma = 30.96 s_ff is resolution-independent over Delta x in [2e-5, 8e-5] m is stated but not shown in any figure or table; please include the supporting data or a convergence plot.","section":"Section 3.1"},{"comment":"The main text should state how many droplets were excluded by the 10% volume filter and whether exclusions are concentrated at particular s_fs/s_ff values.","section":"Figures 10 and 11"},{"comment":"The comparison with Dupuis and Yeomans (2005) is qualitative; please state this explicitly and describe any quantitative metrics used.","section":"Section 4.1, Figure 13"},{"comment":"The reference list contains a rendering issue ('M¨ uller') that should be corrected to 'Müller'.","section":"References"},{"comment":"Equation (12) is introduced as an isosurface of the SPH-interpolated density, but the text does not specify the smoothing length used for the interpolation grid in the marching cubes step; a sentence clarifying this would help reproducibility.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the code availability is commendable. The main risk is that the contact-angle calibration is presented as validation while being partly self-referential; I would support publication after the calibration is independently corroborated or reframed and the isosurface sensitivity is quantified. The Laplace-pressure and oscillating-droplet results are solid and should be highlighted in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth a serious referee, but with one caveat you should know before reading: the surface-tension side is solid, the contact-angle side is a calibration that leans on a shaky isosurface. The new polynomial force profiles (Eq. 8) with the H-scaled prefactor (Eq. 7) are a genuine departure from the Gaussian/cosine/SPH-kernel profiles in earlier PF-SPH work, and the units argument for s in N/m is clean. The Laplace-pressure and oscillating-droplet tests independently give the same linear sigma = 30.96 s_ff, across resolutions, so that central claim holds up.\n\nThe soft spot is the contact-angle calibration. The paper fits simulated droplet shapes to Young-Laplace profiles to measure theta_CA, then uses that same theta_CA to claim agreement with Young-Laplace—partly self-referential, as the reader's report notes. More concerning is the isosurface. Equation (12) sums only fluid particles, but the SPH density evolved in the simulation includes fixed solid particles in the sums (Section 2.2.4). Near the substrate the fluid-only sum is truncated, so the half-density surface shifts by an O(Delta x) offset that is never calibrated. The volume-exclusion filter in Figures 10 and 11 discards droplets whose isosurface volume is off by more than 10%, which is direct evidence the isosurface can be significantly wrong. Since the same criterion is used to measure the contact-angle transition in Figure 14 and the wheat-leaf contact line in Figure 16, any systematic bias near the substrate propagates into those case-study conclusions. That said, the surface-tension validation is independent of this, and the case studies are demonstrations, not quantitative predictions. The hydrophilic-stripe case also misses its target 45° angle, landing at 55°, which the authors acknowledge.\n\nThe stress-test note holds up on reading. The inconsistency between the fluid-only isosurface and the full density field is real, and the 10% filter doesn't establish that the surviving cases are unbiased. Still, the paper is honest about its limitations, and the method is clearly useful for qualitative predictions on complex surfaces. The code is on GitHub, though not pinned with a commit hash.\n\nFor a reader in SPH or wetting simulation, this is a useful contribution. It deserves a serious referee, and I'd bring it to our reading group. Recommendation: send it to peer review, with a request to address the isosurface inconsistency or at least to show that the calibrated contact-angle curve is insensitive to the interface definition.","headline":"New polynomial PF-SPH force profiles give a genuinely scale-independent surface tension calibration, but the contact-angle side leans on a suspect density isosurface and a partly self-referential fit—worth peer review all the same.","tokens_in":20181,"tokens_out":2762,"would_cite":true,"duration_ms":26938,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76M28","76D45"],"pacs":["47.55.D-","47.11.-j"],"model":"deepseek-v4-flash","headline":"A new polynomial pairwise-force profile for smoothed particle hydrodynamics makes simulated surface tension and contact angle scale-independent, so equilibrium sessile droplet shapes can be computed on arbitrary rough and chemically…","keywords":["smoothed particle hydrodynamics","pairwise force model","surface tension calibration","contact angle","sessile droplet","wetting","rough surface","chemically patterned surface"],"falsifier":"Run the calibrated model at particle widths smaller than $2\\times 10^{-5}$ m or larger than $8\\times 10^{-5}$ m on a flat surface, measure the Laplace-pressure slope $\\sigma = 30.96 s_{\\mathrm{ff}}$ and the fitted Young-Laplace contact angle for a fixed $s_{\\mathrm{fs}}/s_{\\mathrm{ff}}$, and check whether both stay on the calibration curves; any drift with resolution would falsify the scale-independence claim. A second check is to compare the volume enclosed by the density isosurface with the known droplet volume near the contact line, since the paper already discards droplets whose isosurface volume is off by more than 10 percent.","tokens_in":19115,"feed_emoji":"💧","tokens_out":14264,"duration_ms":123621,"temperature":0.7,"pith_summary":"This paper tries to establish that a smoothed particle hydrodynamics (SPH) model with a new, physically motivated pairwise force profile can compute equilibrium sessile droplet shapes on arbitrary rough and chemically heterogeneous surfaces without ever tracking the liquid-gas interface or the contact line. The central difficulty it addresses is that once the surface is not flat and axisymmetric, both the shape of the contact line and the contact angle at each point are unknown, so grid-based methods that enforce a prescribed angle lose their footing. The paper's proposed fix is a scale-independent pairwise force: the fluid-fluid interaction strength $s_{\\mathrm{ff}}$ sets the surface tension through $\\sigma = 30.96 s_{\\mathrm{ff}}$, and the ratio of fluid-solid to fluid-fluid strengths $s_{\\mathrm{fs}}/s_{\\mathrm{ff}}$ sets the equilibrium contact angle through a calibrated curve obtained by fitting whole droplet shapes to Young-Laplace solutions. If the claim is right, droplet shapes on leaves, pillared surfaces, and chemically patterned substrates can be simulated with the contact angle emerging from the physics rather than being imposed as a boundary condition, which matters for spray retention, microfluidics, and wetting studies.","feed_headline":"A new droplet model computes shapes on any rough surface","feed_subtitle":"Surface tension and contact angle become two dials, letting rough leaves and patterned chips be simulated directly.","key_machinery":"The load-bearing object is the pairwise force term $\\mathbf{F}^{(\\mathrm{pf})}_i = (H/m_i) \\sum_j s_{ij} f_{ij}(\\|\\mathbf{x}_{ij}\\|/H) \\, \\mathbf{x}_{ij}/\\|\\mathbf{x}_{ij}\\|$, where $f_{ij}$ is a quartic polynomial force profile (equation 8) with a repulsive core, an attractive tail, and a zero crossing placed at $1.1\\Delta x$ for fluid-fluid pairs and $2\\Delta x$ for fluid-solid pairs. The quartic is the lowest-degree polynomial satisfying the endpoint and smoothness constraints, and the fluid-fluid zero crossing is motivated by the nearest-neighbour distance in an optimally packed particle arrangement. The explicit factor $H$ outside the sum makes $s_{ij}$ have the units of an interfacial tension, which is what renders the simulated surface tension independent of resolution. On top of that force, the calibration machinery consists of the virial-pressure measurement of Laplace pressure, the droplet-oscillation frequency, and whole-shape Young-Laplace fitting to the density isosurface $\\langle \\rho(\\mathbf{x}) \\rangle = \\rho_0/2$, whose vertices give the contact angle.","core_discovery":"The central discovery is that the pairwise force profiles chosen in equation (8), together with the $H$-scaled force term in equation (7), remove the resolution dependence that has plagued earlier PF-SPH formulations and turn the simulated surface tension and contact angle into controllable, calibration-ready quantities. In the model, surface tension is produced by fluid-fluid cohesive forces and wetting by fluid-solid adhesive forces, both implemented as short-range repulsive-attractive polynomial forces between particles; the prefactor $H$ in the force term gives the interaction strength the units of N/m, so the resulting surface tension does not change when the particle spacing is refined. The paper calibrates $\\sigma = 30.96 s_{\\mathrm{ff}}$ by measuring Laplace pressure in static spherical droplets and by matching the Rayleigh oscillation frequency of perturbed droplets, and it calibrates the contact angle by fitting semi-analytical Young-Laplace profiles to the density isosurface of simulated sessile droplets. Across different surface tensions and volumes the measured contact angles collapse onto a single curve in the ratio $s_{\\mathrm{fs}}/s_{\\mathrm{ff}}$, supporting the model's scale independence. The same machinery then produces sessile droplet shapes on a square-pillared superhydrophobic surface, on a hydrophobic surface with a hydrophilic stripe, and on a reconstructed wheat leaf with roughness and hairs.","pith_inferences":["If the scale independence extends below the tested particle widths, the same calibration curve could predict surface tension and contact angle for microfluidic droplets without re-fitting; the paper only demonstrates the linear law between $2\\times 10^{-5}$ m and $8\\times 10^{-5}$ m.","The whole-shape Young-Laplace fitting used for calibration could serve as a standard benchmark for comparing other pairwise force profiles, since the paper notes the literature lacks consensus without performing that comparison itself.","The paper's own 10 percent volume filter suggests the density isosurface can mislocate the interface; a more accurate interface reconstruction near the contact line would likely sharpen the contact-angle calibration and the wheat-leaf contact-line visualisation.","The wheat-leaf contact line could be fed directly into an evaporation or retention model as an initial condition; the paper lists this as future work rather than a demonstrated capability."],"forward_implications":["Surface tension can be prescribed directly as $s_{\\mathrm{ff}} = \\sigma/30.96$ at any tested resolution, replacing per-resolution tuning with a single linear calibration.","Equilibrium contact angle can be set by interpolating the ratio $s_{\\mathrm{fs}}/s_{\\mathrm{ff}}$ from the calibrated curve, with the angle emerging naturally at the contact line rather than being imposed.","The model reproduces the suspended-to-collapsed wetting-state transition on square-pillared surfaces when the droplet has a small impact velocity, matching the qualitative behaviour of the experimental study it follows.","Chemically patterned substrates are handled by simply varying the adhesive strength $s_{\\mathrm{fs}}$ across the surface, as demonstrated by the smooth contact-angle transition around a hydrophilic stripe.","Because no interface tracking is required, the same code runs on physically rough and chemically heterogeneous surfaces such as the reconstructed wheat leaf, giving a contact-line shape that would otherwise require a separate free-boundary calculation."],"supporting_citations":[{"why":"Preceding PF-SPH formulation with analytical surface-tension predictions; its scale dependence motivates the new H-scaled term.","marker":"Tartakovsky and Panchenko (2016)"},{"why":"Used PF-SPH for droplet and film flow on rough fracture surfaces, establishing the rough-surface application and an earlier force profile.","marker":"Kordilla et al. (2013)"},{"why":"Studied how surface roughness affects contact angle and droplet flow with PF-SPH, an earlier model the present profiles replace.","marker":"Shigorina et al. (2017)"},{"why":"Introduced the virial-pressure route to surface tension in SPH and earlier pairwise contact-angle modeling, which the calibration uses directly.","marker":"Tartakovsky and Meakin (2005)"},{"why":"Supplies the inviscid droplet-oscillation frequency used to validate the dynamic surface tension.","marker":"Rayleigh (1879)"},{"why":"Provides the semi-analytical axisymmetric Young-Laplace droplet shapes used as the target in contact-angle fitting.","marker":"Danov et al. (2016)"},{"why":"Source of the Young-Laplace ordinary differential equations whose solutions generate the fitted droplet profiles.","marker":"Hartland and Hartley (1976)"},{"why":"Experimental and numerical study of suspended versus collapsed droplets on square pillars that the first case study reproduces.","marker":"Dupuis and Yeomans (2005)"}],"fun_headline_variants":["Calibratable contact angle and tension in new SPH droplet model","Resolution-free droplet shapes on complex surfaces with SPH","SPH droplet model handles rough and chemically patterned surfaces","Tunable surface tension and wetting in SPH for any substrate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every contact-angle measurement and case-study conclusion assumes that the density isosurface at half the reference density, $\\langle \\rho(\\mathbf{x}) \\rangle = \\rho_0/2$, reliably marks the liquid-gas interface and that the contact line lies exactly two particle widths ($2\\Delta x$) from the substrate, so a systematic bias of that isosurface near the contact line would shift the calibrated $s_{\\mathrm{fs}}/s_{\\mathrm{ff}}$-to-angle curve and the computed droplet shapes.","fun_headline_variants_meta":{"raw":{"variants":["Calibratable contact angle and tension in new SPH droplet model","Resolution-free droplet shapes on complex surfaces with SPH","SPH droplet model handles rough and chemically patterned surfaces","Tunable surface tension and wetting in SPH for any substrate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000729,"raw_usage":{"total_tokens":3314,"prompt_tokens":1044,"completion_tokens":2270,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":660,"completion_tokens_details":{"reasoning_tokens":2200}},"tokens_in":660,"tokens_out":2270,"duration_ms":18475,"temperature":1.0,"reasoning_tokens":2200,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:04:39.236777+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the calibrated model at particle widths smaller than $2\\times 10^{-5}$ m or larger than $8\\times 10^{-5}$ m on a flat surface, measure the Laplace-pressure slope $\\sigma = 30.96 s_{\\mathrm{ff}}$ and the fitted Young-Laplace contact angle for a fixed $s_{\\mathrm{fs}}/s_{\\mathrm{ff}}$, and check whether both stay on the calibration curves; any drift with resolution would falsify the scale-independence claim. A second check is to compare the volume enclosed by the density isosurface with the known droplet volume near the contact line, since the paper already discards droplets whose isosurface volume is off by more than 10 percent.","supporting_citations":[{"cited_title":"Pairwise force smoothed particle hydrodynamics model for multiphase flow: Surface tension and contact line dynamics","cited_arxiv_id":null,"evidence_quote":"Preceding PF-SPH formulation with analytical surface-tension predictions; its scale dependence motivates the new H-scaled term."},{"cited_title":"A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces","cited_arxiv_id":null,"evidence_quote":"Used PF-SPH for droplet and film flow on rough fracture surfaces, establishing the rough-surface application and an earlier force profile."},{"cited_title":"Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow","cited_arxiv_id":null,"evidence_quote":"Studied how surface roughness affects contact angle and droplet flow with PF-SPH, an earlier model the present profiles replace."},{"cited_title":"Modeling of surface tension and contact angles with smoothed particle hydrodynamics","cited_arxiv_id":null,"evidence_quote":"Introduced the virial-pressure route to surface tension in SPH and earlier pairwise contact-angle modeling, which the calibration uses directly."},{"cited_title":"Shape analysis of a rotating axisymmetric drop in gravitational field: Comparison of numerical schemes for real-time data processing","cited_arxiv_id":null,"evidence_quote":"Provides the semi-analytical axisymmetric Young-Laplace droplet shapes used as the target in contact-angle fitting."},{"cited_title":"Axisymmetric Fluid-Liquid Interfaces: Tables Giving the Shape of Sessile and Pendant Drops and External Menisci, with Examples of Their Use","cited_arxiv_id":null,"evidence_quote":"Source of the Young-Laplace ordinary differential equations whose solutions generate the fitted droplet profiles."},{"cited_title":"Modeling droplets on superhydrophobic surfaces: Equilibrium states and transitions","cited_arxiv_id":null,"evidence_quote":"Experimental and numerical study of suspended versus collapsed droplets on square pillars that the first case study reproduces."}],"review_version":1}